Large-time behavior of solutions of parabolic equations on the real line with convergent initial data III: unstable limit at infinity
Large-time behavior of solutions of parabolic equations on the real line with convergent initial data III: unstable limit at infinity
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DOI：
10.1007/s42985-022-00187-y
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发表时间：
2021-12
期刊：
影响因子：
--
通讯作者：
Antoine Pauthier;P. Polácik
中科院分区：
文献类型：
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作者：
Antoine Pauthier;P. Polácik
本文是我们研究半线性抛物方程在真实的直线上的有界解的继续和结论。我们假设f是满足次非退化条件的局部Lipschitz函数。我们的目标是描述ofu（x，t）的渐近行为为。在本系列的前两部分中，我们主要考虑的是;或和;或，和是方程的稳定平衡点的情况。在所有这些情况下，我们证明了相应的解是拟收敛的--如果有界的话--也就是说，所有的极限轮廓都是定态的。极限曲线或累积点被纳入。在本文中，我们假设，，和是方程的不稳定平衡点。我们以前的拟收敛定理在这种情况下涉及一些限制性的技术条件的解决方案，我们现在删除。我们唯一的条件是它在某些情况下是非振荡的（只有1000个临界点）。由于已知振荡有界解不总是拟收敛的，所以我们的结果是接近最优的。
This is a continuation, and conclusion, of our study of bounded solutionsuof the semilinear parabolic equationon the real line whose initial datahave finite limitsas. We assume thatfis a locally Lipschitz function onsatisfying minor nondegeneracy conditions. Our goal is to describe the asymptotic behavior ofu(x,t) as. In the first two parts of this series we mainly considered the cases where either; orand; or else,, andis a stable equilibrium of the equation. In all these cases we proved that the corresponding solutionuis quasiconvergent—if bounded—which is to say that all limit profiles ofasare steady states. The limit profiles, or accumulation points, are taken in. In the present paper, we take on the case that,, andis an unstable equilibrium of the equation. Our earlier quasiconvergence theorem in this case involved some restrictive technical conditions on the solution, which we now remove. Our sole condition onis that it is nonoscillatory (has only finitely many critical points) at some. Since it is known that oscillatory bounded solutions are not always quasiconvergent, our result is nearly optimal.